Scattering Theory and Correlation Functions in Statistical Models with a Line of Defect

Abstract

The scattering theory of the integrable statistical models can be generalized to the case of systems with extended lines of defect. This is done by adding the reflection and transmission amplitudes for the interactions with the line of inhomegeneity to the scattering amplitudes in the bulk. The factorization condition for the new amplitudes gives rise to a set of Reflection-Transmission equations. The solutions of these equations in the case of diagonal $S$-matrix in the bulk are only those with $S =\pm 1$. The choice $S=-1$ corresponds to the Ising model. We compute the exact expressions of the transmission and reflection amplitudes relative to the interaction of the Majorana fermion of the Ising model with the defect. These amplitudes present a weak-strong duality in the coupling constant, the self-dual points being the special values where the defect line acts as a reflecting surface. We also discuss the bosonic case $S=1$ which presents instability properties and resonance states. Multi-defect systems which may give rise to a band structure are also considered. The exact expressions of correlation functions is obtained in terms of Form Factors of the bulk theory and matrix elements of the defect operator.